the empirical ft
DESCRIPTION
The Empirical FT. What is the large sample distribution of the EFT?. The complex normal. Theorem. Suppose p.p. N is stationary mixing, then. Proof. Write. Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments. Consider. - PowerPoint PPT PresentationTRANSCRIPT
The Empirical FT.
)()0( Note
real ),(}exp{}exp{)(:
}...{0
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tdNtiidEFT
TData
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N T
j
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N
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What is the large sample distribution of the EFT?
The complex normal.
2var
)1,0( ...
onentialexp2/2/)(|)1,0(|
2/)()2/1,0()1,0(
||var
:Notes
)2/,(Im ),2/,(Ret independen are V and Uwhere
V i U Y
form theof variatea is ),,( normal,complex The
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INZZZ
ZZN
iZZINN
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NN
N
j
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Theorem. Suppose p.p. N is stationary mixing, then
))(2,0( asymp are
~/2 with 0 integersdistinct ,..., ),2
(),...,2
( ).
)(2,0(
asymp are 0 anddistinct ,..., ),(),...,( ).
0 )),(2,0(
0 )),0(2,(
allyasymptotic is )( ).
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L11
NN
C
lLLTT
lNN
C
L
T
N
T
N
NN
C
NNN
T
N
TfIN
TrrrT
rd
Tr
diii
TfIN
ddii
TfN
TfTpN
di
Evaluate first and second-order cumulants
Bound higher cumulants
Normal is determined by its moments
Proof. Write
)()()( N
TT
N dZd
Consider
)(2)()(
saw We
)()(2~
)()()(
)()()()()}(),(cov{
TTT
NN
T
NN
TT
NN
TTT
N
T
N
d
f
df
ddfdd
Comments.
Already used to study rate estimate
Tapering makes
dfHd NN
TT
N )(|)(|~)(var
Get asymp independence for different frequencies
The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0
Also get asymp independence if consider separate stretches
p-vector version involves p by p spectral density matrix fNN( )
Estimation of the (power) spectrum.
22
2
2
2
2
2
2
2
)(}2/)(var{ and
)(}2/)({ notebut
0)( unlessnt inconsiste appears Estimate
lexponentia ,2/)(~
|))(2,0(|2
1~
|)(|2
1)(
m,periodogra heconsider t ,0For
NNNN
NNNN
NN
NN
NN
C
T
N
T
NN
ff
ffE
f
f
TfNT
dT
I
An estimate whose limit is a random variable
Some moments.
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22
|)(|)(|)(||)(|
)(}exp{)(
|| var||
r.v.Complex
T
NNN
TT
N
T T
NN
T
N
pdfdE
so
pdtptiEd
EE
The estimate is asymptotically unbiased
Final term drops out if = 2r/T 0
Best to correct for mean, work with
)()( TT
N
T
N pd
Periodogram values are asymptotically independent since dT values are -
independent exponentials
Use to form estimates
sL' severalTry variance.controlCan
/)(}2/)(var{ )(}2/)({
Now
ondistributiin 2/)()(
gives CLT
/)/2()(
Estimate
near /2
0 integersdistinct ,...,Consider .
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2
2
2
2
2
1
LfLffLfE
Lff
LTrIf
Tr
rrmperiodograSmoothed
NNLNNNNLNN
LNN
T
NN
ll
TT
NN
l
L
Approximate marginal confidence intervals
LVT
L
ff
fT
T
data,split might
FT or taperedmean, weightedmight take
estimate consistentfor might take
ondistributi valueextreme viaband ussimultaneo
levelmean about CIset
logby stabilized variance
Notes.
)}2/(/log)(log)(log
)2/1(/log)(Pr{log2
2
More on choice of L
biased constant,not is If
radians /2
(.) of width affects L of Choice
)()(W
)2/)/(()2/)(sin2
11
/2/2 },/)({)}({
Consider
T
2
T
T
T
lll
lll
l
T
ff
TL
W
df
dfTTL
TrTrLIEfE
Approximation to bias
0)( symmetricFor W
...2/)()(")()(')()(
...]2/)(")(')()[(
)()(
)()(
)()( Suppose
22
22
11
dW
dWfBdWfBdWf
dfBfBfW
BfW
dfW
BWBW
TTT
TT
T
T
TT
T
Indirect estimate
duuquwuip T
NN
TT
N )()(}exp{21
2
Could approximate p.p. by 0-1 time series and use t.s. estimate
choice of cells?
Estimation of finite dimensional .
approximate likelihood (assuming IT values independent exponentials)
)}/2(/)/2(exp{)/2()(
in ),;( spectrum
1 TrfTrITrfL
f
r
T
Bivariate case.
)( )(
)( )(
matrixdensity spectral
)()()}(),(cov{
)(
)(/]1}[exp{
)(
)(
NNNM
MNMM
MNNM
N
M
ff
ff
dfdZdZ
dZ
dZitit
tN
tM
Crossperiodogram.
T
NN
T
NM
T
MN
T
MMT
T
N
T
M
T
MN
II
II
ddT
I
)( formmatrix
)()(2
1)(
I
Smoothed periodogram.
LlTrLTr ll
l
TT
NN ,...,1 ,/2 ,/)/2()( If
Complex Wishart
ΣW
XXWΣ
0XX
nE
nW
IN
n T
jj
C
r
rn
squared-chi diagonals
~),(
),(~,...,
1
1
Predicting N via M
)()(1}exp{
)(
}exp{)()2()(
)()(/)()( :coherency
)(]|)(|1[ :MSE
phase :)(arggain |:)(|
functionfer trans,)()()(
|)(-)(|min
)(by )( predicting
1
2
1
2
M
NNMMNM
NN
MMNM
MNA
MN
dZAiit
tN
diuAua
fffR
fR
AA
ffA
AdZdZE
dZdZ
Plug in estimates.
LRE
R
LL
RRLLFR
R
fffR
fR
AA
ffA
T
TL
T
NNMMNM
T
T
NN
T
TT
T
MM
T
NM
T
/1||
)-(1-1
point %100approx 0|| If
)1()1()(
)||||;1;,()||1(
|| ofDensity
)()(/)()(
)(]|)(|1[ :MSE
)(arg |)(|
)()()(
2
1)-1/(L
2
22
122
2
2
1
Large sample distributions.
var log|AT| [|R|-2 -1]/L
var argAT [|R|-2 -1]/L
Networks.
partial spectra - trivariate (M,N,O)
Remove linear time invariant effect of M from N and O and examine coherency of residuals,
)(
)()()()()(
M of effects
invariant lineartime removed having O and N of spectrum-cross partial
),()()()(
),()()()(
|
1
|
1
|
1
|
T
MNO
MNMMNMNOMNO
MMOMMOMO
MMNMMNMN
R
fffff
ffBdZBdZdZ
ffAdZAdZdZ
Point process system.
An operation carrying a point process, M, into another, N
N = S[M]
S = {input,mapping, output}
Pr{dN(t)=1|M} = { + a(t-u)dM(u)}dt
System identification: determining the charcteristics of a system from inputs and corresponding outputs
{M(t),N(t);0 t<T}
Like regression vs. bivariate
Realizable case.
a(u) = 0 u<0
A() is of special form
e.g. N(t) = M(t-)
arg A( ) = -
Advantages of frequency domain approach.
techniques formany stationary processes look the same
approximate i.i.d sample values
assessing models (character of departure)
time varying variant
...
Published by AAAS
R. Fuentes et al., Science 323, 1578 -1582 (2009)
Fig. 2. DCS restores locomotion and desynchronizes corticostriatal activity
Muscle spindle example